A REMARK ON WEIGHTED REPRESENTATION FUNCTIONS

Let $G$ be a finite abelian group, and $k_1,k_2$ be two integers. For any subset $A\subset G$, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we generalize a result of Q.-H. Yang and Y.-G. Chen to finite abelian groups. More precisely, we characterize all subsets $A\subset G$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(G\backslash A,n)$ for all $n\in G$.